Twelve tone harmony abandons the cherished terms associated with tonality. First to go is key. Twelve tone harmony has no key. Twelve tone harmony uses a single scale, the chromatic scale. Next to go is chord. Twelve tone harmony does not do chords, it does collections of pitches. Chord implies tonality, a collection of pitches implies atonality.
Twelve tone harmony is mathematical in nature. Luckily the maths is straightforward and there are only a few big concepts:
A set is a collection of things. A set is denoted by curly brackets {}.
A pitch class is a set of pitches. The pitch class {0,1,2,3,4,5,6,7,8,9,10,11} contains the 12 notes in the chromatic scale numbered from 0 to 11.
A tone row is a pitch class with up to 12 notes. The notes are chosen at random from the chromatic scale and no note is repeated. {0,8,5,4,11,2,7,1,9,6,3,10} is an example of a 12 tone row. {0, 4, 7} is an example of a 3 tone row (and represents a major triad).
A tone matrix is a list of tone rows in the form of a matrix, a square grid. A 3 tone matrix contains 3 tone rows and 9 notes, a 12 tone matrix contains 12 tone rows and 144 notes.
Figure: twelve tone harmony matrix
The twelve tone harmony matrix figure shows the steps involved in designing a tone matrix. Writing twelve tone harmony involves designing a tone matrix. The following example shows how to construct a 3 tone matrix. Exactly the same procedure is used to construct a 12 tone matrix.
Draw an empty matrix.
Fill in the top row. Start with the number zero then choose two other numbers at random in the range 1 to 11. The completed tone row, {0,8,5}, constitutes the prime tone row. It is equivalent to a melodic idea.
Invert the prime tone row and place the result in the first column. Inversions always add up to 12 except for
0 which remains unchanged. The invert of the prime tone row {0,8,5} is {0,4,7} because 8+4=12 and 5+7=12.
Transpose the prime tone row and place the result in the second row. The first number of the transposed tone row, 4, is already entered. Add 4 to 8 to get 12, convert to 0, and put it in the second column of the second row. Add 4 to 5 to get 9 and put it in the last column. The tone row {4,0,9} is a transposed variant of the prime tone row {0,8,5}.
Transpose the prime tone row again and place the result in the third row. This time add 7 to the prime row to get another transposed tone row, {7,3,0}.
Two further transformations are derived from the completed matrix:
A retrograde variant is a tone row read from right to left. For example, the prime tone row, {0,8,5} has a retrograde variant {5,8,0}.
An invert retrograde variant is a tone row read from top to bottom. For example, the second row {4,0,9} has an invert retrograde variant, {3,0,8}, the second column read from bottom to top.
A twelve tone matrix provides the raw material for twelve tone harmony. It is up to you how to use it. Randomly choose a tone row from the prime and its variants; play a series of single notes in a tone row; play a sequence of notes in a tone row simultaneously; double any note at the octave above or below; play two or more tone rows simultaneously; offset one tone row to overlap with another. There are all sorts of permutations and combinations to play with.
This is how twelve tone harmony was written:
The pitch class {0,8,5,4,11,2,7,1,9,6,3,10} is the prime tone row. Note C4 (middle C) is note 0. The harmony is in the treble clef.
The bass clef contains an invert retrograde variant of the prime tone row {5,10,1,8,7,4,0,2,9,6,3,11}. Note C3 is note 0.
The invert retrograde variant is offset and starts four beats after the prime tone row.
Both tone rows are played twice. The first time round the notes in each tone row are grouped in twos, the second time they are randomly grouped in ones, twos or threes.
